Abstract
A new definition of spectral data of a monopole is given for any compact Lie or Kac-Moody group. It is shown that the spectral data determines the irreducible monopole. In the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the Dynkin diagram of the group. The structure of solutions to Nahm's equations corresponding to the monopole is discussed.
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Bernstein, I. N., Gel'fand, I. M., Gel'fand, S. I.: Schubert cells and the cohomology of the spaces G/P. Russ. Math. Surv.28(3), 1–26 (1973)
Garland, H., Murray, M. K.: Why instantons are monopoles. Commun. Math. Phys.121, 85–90 (1989)
Gravesen, J.: On the topology of spaces of holomorphic maps. D. Phil Thesis. Oxford. (1987)
Grothendieck, A.: Sur la classification des fibrés holomorphes sur la sphère de Riemann. Am. J. Math.79, 121–138 (1957)
Hitchin, N. J.: Monopoles and geodesics. Commun. Math. Phys.83, 579–602 (1982)
Hitchin, N. J.: On the construction of monopoles. Commun. Math. Phys.89, 145–190 (1983)
Hitchin, N. J.: Murray, M. K.: Spectral curves and the ADHM construction. Commun. Math. Phys.114, 463–474 (1988)
Humphreys, J. E.: Introduction to Lie algebras and representation theory. Berlin, Heidelberg, New York: Springer 1972
Humphreys, J. E.: Linear Algebraic Groups. Berlin, Heidelberg, New York: Springer 1981
Hurtubise, J. C., Murray, M. K.: On the construction of monopoles for the classical groups. Commun. Math. Phys.122, 35–89 (1989)
Kobayashi, I. N., Nomizu, K.: Foundations of differential geometry, Vol. 1. New York: Interscience, Wiley 1969
Murray, M. K.: Non-abelian magnetic monopoles. Commun. Math. Phys.96, 539–565 (1984)
Murray, M. K.: Stratifying monopoles and rational maps. Commun. Math. Phys.125, 661–674 (1989)
Nahm, W.: Self-dual monopoles and calorons. In: Group Theoretical Methods in Physics. Denado, G. et al. (eds), Trieste. Lecture Notes in Physics vol201. Berlin, Heidelberg, New York: Springer 1983
Pressley, A., Segal, G.: Loop groups. Oxford: Oxford University Press 1986
Tennison, B. R.: Sheaf Theory. Lond. Math. Soc. Lecture Note Series20. London, New York, Melbourne: Cambridge University Press 1975
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Communicated by A. Jaffe
Research supported in part by NSER C grant A8361 and FCAR grant EQ3518
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Hurtubise, J., Murray, M.K. Monopoles and their spectral data. Commun.Math. Phys. 133, 487–508 (1990). https://doi.org/10.1007/BF02097006
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DOI: https://doi.org/10.1007/BF02097006